On the optical constants of metals at wavelengths shorter that their critical wavelengths

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On the optical constants of metals at wavelengths shorter that their critical wavelengths

W.R. Hunter

To cite this version:

W.R. Hunter. On the optical constants of metals at wavelengths shorter that their critical wave-

lengths. Journal de Physique, 1964, 25 (1-2), pp.154-160. �10.1051/jphys:01964002501-2015400�. �jpa-

00205728�

ON THE OPTICAL CONSTANTS OF METALS

AT WAVELENGTHS SHORTER THAT THEIR CRITICAL WAVELENGTHS (1) By W. ^{R. HUNTER,}

E. O. Hulburt Center for Space Research, U. S. Naval Research Laboratory, Washington, ^{D. C.}

Résumé.

^{On a} étudié les propriétés optiques ^{de Al,} In, Mg ^et Si dans l’extrême ultra-violet pour des longueurs ^d’onde inférieures à la longueur ^d’onde critique. ^{On a} trouvé que Al, Mg ^{et Si,} qui ont des électrons de valence faiblement liés et des électrons internes fortement liés, peuvent

être décrits, ^{avec une bonne} approximation, par la théorie des électrons libres. Pour l’indium, dans ce domaine spectral, le terme dominant de la constante diélectrique complexe est le terme dû aux électrons libres ; cependant, d’autres mécanismes d’absorption ^ne sont pas négligeables,

aussi on ne peut pas utiliser ^une simple théorie à deux paramètres.

Abstract.

^The optical properties of Al, In, Mg, and Si have been investigated ⁱⁿ the extreme ultraviolet at wavelengths ^shorter than their critical wavelengths. ^{It was} found that Al, Mg,

and Si, which have loosely ^bound valence electrons and tightly ^{bound core electrons can be des-} cribed, ^{to a} good approximation, by the free electron theory. ^For In, ^{in this} spectral ^{range, the}

dominant term in the complex dielectric constant is the free electron term, however, ^other absorp-

tion mechanisms are not negligible ^{so a} simple two-parameter thoery cannot be used.

Introduction.

One of the consequences of Maxwell’s theory ^{of the} propagation of electro-

magnetic ^{waves in a medium} containing ^free charges ^{is the existence of a critical} wavelength Àc.

Wavelengths greater ^{than are} reflected while the shorter wavelengths penetrate ^{into the medium.}

A simple explanation ^{of Xc is that the free charges}

oscillate ⁱⁿ phase ^{with the} electromagnetic ^waves

for À > Xc but ^{are 7r} radians out of phase ^for

À Àc. ^{Thus the} energy reradiated by ^{the oscil-} lating charges interferes constructively ^{with the}

transmitted ^wave and destructively ^{with the}

reflected ^{wave for À} Ào and ^{vice versa.}

In the case of a metal, ^{the free} charges ^are

electrons which, by ^{virtue of} their coulomb inter-

actions, ^are capable ^of collective oscillations. Con-

sequently ^{it is to} be expected that metals will have

a Xc given by ;

where c ^{is the} velocity ^of light, e is the electronic

charge ^{and N} the valence electron density.

This simple ^{model was used} by ^Zener [1] ^to explain ^the experimental ^{results of Wood [2] who}

showed that the alkali metals are transparent ⁱⁿ

the ultraviolet. This theory ^was successful in that it gave fair approximations ^{to the observed wave-} length ^at which transmission set in. However, ^it requires ^total reflection at all angles of incidence for X > Àc which ^{is not the case.} Kronig [3] ^modi-

fied Zener’s theory by introducing damping ^{of the}

free electron motion, ^{due to} collisions with the (1) Supported, ^{in part, by} the National Aeronautics and

Space Administration.

lattice. This then becomes the free electron theory.

of Drude [4] and, for the purpose of describing optical phenomena, ^{can be} put ^{into the} following

form :

In the limit as 1:" approaches infinity, ^eq. (1)

becomes :

and _eq. (2) ^becomes zero, i.e., ^{Zener’s model.} el

and _e2 are the real and imaginary parts ^{of the-e} complex dielectric constant, respectively ; n ^{is the}

index of refraction, ^{k is} the extinction coefficient,

Me is the critical angular frequency ^{and r is the}

relaxation time of the electrons.

A Drude type model describes fairly ^{well the} optical properties ^{of the alkali} metals which have

a loosely bond valence electron and tightly ^bound

core electrons [5]. Hence, ^{it is} reasonable to expect

that the theory ^{would be at least a} good ^first approximation ^{to other} metals that have loosely

bound valence electrons and tightly ^{bound cores.}

Pines [6] ^{discussed the} classification of metals by

the binding energy of the valence and core electrons and found that the metals listed in Table I have this characteristic in common with the alkali metals. Thus the inference can be made that a

critical wavelength exists for these metals at which transmission should commence. In Table I is

listed, ^{for several} elements, the value for ae ^and the nearest absorption edge ^{at which k should once}

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01964002501-2015400

155

TABLE I

again ^become large enough ^{to reduce or} prevent

transmission.

Metals that cannot be described by. ^{the free}

electron theory may still have ^a X, accompanied by

transmission for À X,. ^{This means that the free} electron term in the complex dielectric constant is dominant in the region ^{around Xc,} however, ^other absorption processes will ^not be far enough ^remo-

ved from the region ^to be negligible. ^Some examples ^are indium, tin, antimony, and bismuth.

Other metals, ^{such as} gold [7], ^{have no} optical

transmission and, therefore, ^{no trace of a xc,} although ^{it can be} predicted ^on the basis of theory.

If the metal becomes transparent ^{for X} Xc, k must be rather small. Furthermore, ^{from a}

consideration of eq. (3), n ^will be real but less than

unity, ^so that radiation can center the medium

only ^at angles less than the critical angle, ^arc.

Hence, ^{for X Àc} metals that have transmission bands or regions ^{will be} expected ^{to have a} very small k and n less than unity.

Experimental ^methods.

The metals to be

investigated ^were in the form of thin evaporated films, deposited ^under optimum conditions as des- cribed by Hass, Hunter, ^and Tousey [8]. ^{In the}

case of unbacked films, ^{a suitable} parting agent

was placed ^{on the substrate} prior ^to the evapora- tion of the metal, ^{and another} substrate, ^situated alongside, ^was exposed during ^the evaporation ^and

was subsequently ^coated with silver and used to

measure the film thickness by multiple ^{beam inter-} ferometry. Details of the process of making

unbacked films will be published [9].

The extinction coefficient, k, ^{was measured} by

transmission through unbacked metal films. By measuring the transmittance of films of different

thicknesses, absorption ^at the surface by ^{an oxide} layer, and also reflection losses, ^{can be} eliminated,

assuming they ^{are the same in both cases. The}

formula for obtaining ^k by ^{this method is :}

where x is the film thickness and T the transmit- tance.

If k is very small, ^{and n} is less than unity, ^{it is} possible ^{to use a} technique ^{similar to} the critical

angle ^{method for} measuring ^{the index of non-} absorbing ^{solids and} liquids ^{in the visible} region.

Calculations of reflectance versus angle ^{of inci-}

dence for a medium with n

0. 707, 0152c 450 and

FIG. 1.

Calculated reflectance versus angle ^{of incidence}

for n

0.707, ac £ 450, and different values of k.

different ^{values of} k, ^{shown in} figure 1, ^illustrate

that a definite «Q ^can be observed only ^{if k}

0, however, ^{for small} values of k there is ^a sudden

drop ⁱⁿ reflectance in the region ^{of (Xc. The} index,

n, ^was obtained from the position ^{of maximum} slope ^of the reflectance versus angle ^{of incidence}

curve. The position ^{of the} angle ^{of maximum} slope, ^{CXm, of the} reflectance curves is very close Lo ao, but moves further away as k increases.

Eventually ^{k becomes} large enough ^{so that the}

R vs oc curve is monotonic, ^{as is the case of k} > 0. 2.

Further calculations were made to find the

dependence ^{of am on} k. The results are shown in

figure 2, ^{where the} angle ^{of incidence of am is}

shown as a function of k for different values of the

FIG. 2.

Calculated position ^{of the} angle of maximum

slope, ^{am, as a} function of k. The full lines are for unpo- larized radiation, the dotted lines for the p-component

and the dashed lines for the s-component.

index. The curves for n > 0.7 are terminated when the respective ^{R vs oc curves become mono-}

tonic. By ^{means of these curves a} correction for the value of n

sin _«m can be determined if k is known for the material. For example, ^{if the mea-}

sured value of «m

49 . 5°, ^{and k}

^0.2 (fig. 2)

indicates that n

0.6 ; neglect of the correction would yield n

^{0. 76, an error of about 27 %.}

Polarization would cause the measured n to vary between 0.68 for complete s-polarization ^{and 0.81}

for complete p-polarization, corresponding ^{to errors}

of 15 % ^{and 33} %, respectively. ^This is, ^of

course an extreme case ; corrections for k # ^{0 are} usually quite ^small.

The effect of reflections from the metal-substrate interface ahd the oxide layer ^are discussed else- where [10] ^{and were shown to} be, ^{for the most} part, negligible.

Results.

The results for aluminum are shown in figure ^3. In the upper part ^{of the} figure, ^the

dots through ^{which a solid line has been} drawn, represent measured values of sin _«m. The large

shaded circles, ^{at 584} A and 736 A, ^are the results of previous measurements [11] using ^{an interfer-}

FIG. 3.

Optical constants of aluminum at wavelengths

shorter than the critical wavelength.

ence technique. ^{At the same} wavelengths, ^the triangles ^{are data obtained} by Madden, Canfield,

and Hass [12] ^from oxide-free surfaces using ^the general reflectance method. Data obtained by

LaVilla and Mendlowitz [13] ^{from inelastic elec-}

tron scattering experiments ^{are shown} as crosses.

Index measurements could only be carried out to 200 A because, ^{at shorter} wavelengths, ^am > 84o,

which was the largest angle ^{at which} measurements

could be made. The open circles represent n ^cal-

culated from the free electron theory using

Xo

837 A and T = 1. 1 X 10-15 seconds [14],

and are in very good agreement with the measured values. From 700 A to longer wavelengths, ^the

broken curve represents the correction necessary because of the displacement ^{of ocm} by ^{the ever-} increasing ^k.

In the lower half of the figure ^{are shown the}

measured values of k. The scatter of points ^is quite large, especially ^at wavelengths shorter than the L2.3 x-ray edge ^{at 170} A, ^{where the dotted}

line is to be regarded merely ^{as an} indication of the behavior of k. For X > 170 A, ^{the results are}

somewhat more consistent and the scatter of data decreases with increasing wavelength. ^Once again

the large ^{shaded circles} represent ^{the results of} previous measurements by transmission while the

crosses and triangles ^{are data of La Villa and}

Mendlowitz and Madden, Canfield, ^and Hass, ^res-

pectively. The. ^{diamonds represent the data of}

157

Tomboulian and Bedo [15], ^obtained photogra- phically using ^a synchroton ^{as a} light ^source.

Values of k, calculated using the free electron theory,

are so labelled. There is ^a definite departure ^from

the free electron model in that the measured k does not decrease with decreasing wavelength ^as rapidly

as the calculated values. This is due to the fact that interband transitions at approximately ^{8 OOOA}

which are not included in the Drude theory ^are

still exerting ^a slight influence.

The data of Tomboulian and Bedo show a sharp dip in the value of k at the L2,3 edge. ^{Astoin and}

Vodar [16], ^{who have measured} the transmittance of aluminum films have reported ^a sharp peak ⁱⁿ

transmittance at the L2,3 edge, corresponding ^to

the dip ^{in k observed} by Tomboulian and Bedo.

Both sets of investigators used aluminum films backed by ^thin films of zapon ^or collodion. In the earlier work of Tomboulian and Pell [17] ^on

unbacked aluminum films, ^{5 000 A} thick, ^the sharp peak ^was. absent. No sharp peak ^was observed in the present work, although ^{it would have been}

detected in spite ^{of the} large ^{scatter of the} points

near the edge.

In Iigure ^{4 are shown the} optical properties ^of

aluminum from 100 A to 6 000 A. Open ^circles represent ^values taken from the experimental

FIG. 4.

Optical constants and normal incidence reflec- tance of aluminum.

curves of the preceeding figure, otherwise the data

symbols ^{have the same} meaning ^{as in the} previous figure ^{with the} exception of the additional data of Hass and Waylonis [18] ^{for the near} ultraviolet and visible, ^{shown as} plus signs.

.

The solid line, ^drawn through ^{the index} data, represents ^{the index} calculated using the free elec- tron theory ^{with the} parameters given above, ^and

is in very good agreement ^{with the} experimental

data at all wavelengths. ^Good agreement ^{is also} obtained in the visible region between the data

points ^{for k and} the calculated values, ^{which are}

shown by ^{the solid line. At} wavelengths ^shorter

than about 860 A, however, ^{the curve} given by ^the

free electron theory drops ^more rapidly ^{than the} experimental values, ^shown by the open circles,

for the reason given ^{earlier. The} dotted line is the reflectance calculated using ^{the free electron} theory

wich

agrees very well with the measured values from 6 000 to 1 025 A. At shorter wavelengths, ^the

reflectance data points, ^connected by ^{the dashed} line, represent calculations from the measured n and k since interference ^effects present ^{in thin}

films prevent ^direct measurements of reflectance.

The true values of n and k still remain to be found in the interval between 1 000 A and 2 000 A.

Because ^{of the} agreement between calculated and measured reflectances, however, ^{it is} expected ^that

the measurements will not show large ^deviations

from those represented by ^{the solid lines.}

Indium has two distinct spectral regions ^where

the value of k becomes _very small, ^{from 120 A to}

shorter wavelengths, the cut-off wavelength ^could

not be determined ; ^{and from 744 A to} approxi- mately ^{1 100 A.} Apparatus ^{was not available for}

index measurements at the very short wavelengths

so that only measurements of the extinction coef- ficient ^were made. The results are shown in Table II. Because of experimental difficulties, ^the

numbers are accurate only ^{to within 50} %.

TABLE II

The optical ^{constants of indium from 744 A to}

1 085 A are shown in figure 5, where measured values are represented by dots. The broken line shows the index curve after correction for the effect of non-zero k.

Although ^the free electron term ^{in the} complex

dielectric constant is dominant in this region, ^other

terms are not negligible. ^For example, ^the sharp

increase in k, ^which begins approximately ^{at 760} A,

indicates the commencement of an optical absorp-

tion process, thus introducing ^{another term into}

the complex dielectric constant. This corresponds

to the sudden termination of transmission observed

by Walker, Rustgi, ^{and Weissler} [19] ^{and attri-} buted, by ^{them to} interband transitions. Hence,

no simple two-parameter ^{model can be made to fit}

the data.

At wavelengths ^{less than 800} A, the location of

«m was uncertain and the values shown may be in

error by ⁵⁰ % ^{or more. An} attempt ^{was made to}

obtain more accurate values by matching ^{the cal-}

FIG. 5.

Optical constants of indium from 744 A to 1085 A.

culated transmittance ^to that which was observed

by varying ^n, however, ^the calculated transmit- tance was insensitive to the value of n chosen and the attempt ^{was abandoned.}

The critical wavelength associated with the free

Fm, 6.

2nk/(n2 ⁺ k2)2 ^{as a function of} wavelength ^{for n}

aluminum and indium. c

electron term, may be determined from the _quan-

tity, 2nk/(n2 + k2)2 [20, 21] ^which should have

a sharp peak very close ^to the position ^{of Àc. In} figure 6, ^this expression ^{is shown as a function of} wavelength ^{for indium and aluminum. The circles} represent experimental points ^{obtained in the} present experiment, while the crosses, for alumi- num, ^are the data of LaVilla and Mendlowitz.

The solid line for aluminum was calculated using n

and k obtained from the free electron theory, ^while

the line for indium is only to connect the data

points, ^{since no model was available} for compar- ison.

Since, ^{in this} experiment, ^the optical ^constants

of indium were measured at wavelengths ^shorter

than the critical wavelength, ^the peak ^{of the curve}

could not be located experimentally. By ^extra- polating ^the corrected index curve for indium, however, ^a peak ^{was found at 1080} A, ^shown by

the dashed line, ^which agrees very well with the characteristic energy losS- reported by ^Robins [22]

at,1097 Å. Similarly, ^{for aluminum the} optical

data extend only ^{to 800 Å.} However, ^{the data}

of LaVilla and Mendlowitz agree very closely ^with

the calculated curve which has a peak ^at approxi-

mately ^{840 A..}

The occurrence of optical transmission suggests

that ac may be found from the onset in trans- mission. This a rather inaccurate way in which ^to determine Àc ^{since the} thickness of the film and the sensitivity ^{of the} measuring equipment ^essen- tially ^{determine the} longest wavelength ^{at which} transmittance ^occurs. For example, Walker, Rustgi, and Weissler have published transmission

curves for aluminum and indium, among other

metals, ^{and found the onset of} transmission to occur at approximately ^{855 A and 1 100} Å, respec-

tively. ^{In this} experiment, transmission measure-

.ments could be made to wavelengths ^as long ^as

834 and 1085 A on aluminum and indium, respec-

tively, while Wilkinson [23] ^has photographed ^lines

close to 1200 A through indium films.

Mendlowitz [21] ^has pointed ^{out that the onset}

of transmission _may not occur at z ^{but at a some-} what shorter wavelength ^{at which k} becomes small

enough ^to permit transmission. He has also invest-

igated ^{the location of the maximum in the} quant- ity, 2nk/(n2 ⁺ k2)2 ^{and has shown} that, ^{for non-}

zero damping, ^the peak occurs, at ^a wavelength longer ^than Xc. The shift is very small for alumi- num, amounting ^to approximately ^one part ⁱⁿ

5 000, and since the curve for indium is quite similar, ^{it is} expected ^{that the maximum} is very close to Xc.

The results for magnesium ^{are shown in} figure ^7.

In the upper part ^{of the} figure, ^{the dots} through

which the solid line has been drawn represent ^the

measurements of sin _am. The open circles are cal-

culations using the free electron theory ^{with the}

159

FIG. 7.

Optical constants of magnesium ^at wavelengths

shorter than the critical wavelength.

parameters ?,,, = 1198 A and r =1.1 X 10-l’ [14]

The calculated values _agree fairly ^{well with the}

measured values, although ^{not as well as in the}

case of aluminum. The greater departure ^{of the} points from the smooth curve from 600 to 800 Å

is attributed to uncertainties in the measurements.

However, ^{some electron} scattering experiments (24) ^{show a broad} peak ^{centered at about 20} eV,

believed to be due to MgO, ^which may have in-

fluenced the result.

FIG. 8.

Optical ^constants of silicon at wavelengths shorter than the critical wavelength.

The lower part ^{of the} figure ^shows k, ^calculated using ^{the free electron} theory. Kroger ^{and Tom-}

boulian [25] ^have reported ^{values of} k, ^measured using ^a photographic technique, ^{from 230 A to}

775 A. Their values, ^{which were} converted from

absorption ,coefficient _{u, to} extinction coefficient k,

are shown ^as diamonds. The values they reported

are approximately ^{one order of} magnitude larger

than those calculated using the free electron theory.

In view of the departure ^of the measured k values of aluminum from those calculated using ^{the free}

electron theory this behavior does not seem unrea-

sonable. There is also some evidence of structure in their _curve, aside from the L2.3 ^x-ray edge ^at approximately ²⁵⁰ A, that could not be expected

to appear ⁱⁿ calculations using the free electron

theory.

Since measured values of k with which to correct.

the index curve are not available, ^{the calculated}

values have been used. The corrected curve is shown by ^{the broken} line in the upper half ^{on the}

figure.

The results for silicon are shown in figure 8, ^with

dots and open circles, measured and calculated,

FIG. 9.

Normal incidence reflectance of various metals in the extreme ultraviolet. Data points ^A [12] ^and

X [13] ^{are for} aluminum, and 4 [27] ^{is for germanium.}

The solid lines were calculated using ^{a Drude} type ^model

with the following parameters [14] :

respectively. ^The crenulated ^{circles are data of}

Sasaki and Ishiguro [26]. ^At the shorter wave-

lengths the measured and calculated .data _agree rather well ; however, ^from approximately ^{500 A}

to longer wavelengths, the corrected index curve

drops ^more rapidly ^{than the} calculated values.

The corrected curve has the same slope ^{as the data}

of Sasaki and Ishiguro although ^the magnitudes

are not quite ^{the same.}

In the lower part ^{of the} figure ^{are shown the}

calculated values of k using the free electron theory

with the parameters indicated. Since the free electron theory ^{does not} take x-ray edges ^into account, ^{the curve does not show the sudden} increase in k that must exist at the Lz.,3 edge ^and possibly ^{at the} L1 edge ^also. The data of Sasaki and Ishiguro ^{fit the} calculated values approxi- mately ^{as far as} magnitudes ^{are concerned but the} slope ^{is different.}

Reflectance data for some of the metals listed in

’Table I are collected in figure ^{9. The} solid lines

are the reflectance at normal incidence calculated

on the basis on the free electron theory using ^the parameters given ^{in the} figure caption. ^{The data} points indicate that the approximation is close for aluminum and germanium. ^For aluminum, ^the triangles represent ^{the data of} Madden, Canfield,

and Hass, while the crosses are the calculated reflectances from the data of LaVilla and Mendlo- witz. The diamonds represent ^{data tor} germa- nium obtained by Madden [27].

Acknowledgments.

The author - is pleased ^to acknowledge the efforts of D. W. Angel ^{who made}

the unbacked films of aluminum and indium,

S. G. Tilford who discovered the very short ^wave-

length window ⁱⁿ indium, and R. L.. Blake,

J. F. Meekins, ^and A. E. Unzicker who measured the transmittance of the indium films at the very short wavelengths.

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